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The negative sign in Bohr's equation is there because E=0 when the electron and neutron are separated completely (when the electron is free). So, as they get closer (or get lower in orbitals), they are losing energy and therefore the energy calculated when doing Bohr's equation is negative.
Bohr assumed that when an electron is removed completely, the energy of the system is 0. However, when the electron is still around the nucleus, the system is more stable and thus at a lower energy. Thus, there is a negative sign in front of the equation.
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